Square function estimates and local smoothing for Fourier integral operators

We prove a variable coefficient version of the square function estimate Guth–Wang–Zhang. By classical argument Mockenhaupt–Seeger–Sogge, it implies full range sharp local smoothing estimates for 2 + 1 $2+1$ -dimensional Fourier integral operators satisfying cinematic curvature condition. In particular, conjecture wave equations on compact Riemannian surfaces is settled.